False friends are quite common between Spanish and English. These two languages often have words derived from a common etymological stem (usually Latin) but that eventually have evolved to different meanings. Some common examples are "carpet" and "carpeta" (folder) or "to remove" and "remover" (to stir). Even Babelfish translates the latter wrongly!

Confusion is usually harmless, but not always: if you natively speak Spanish and catch a cold while abroad, you might find yourself in trouble if you ask the pharmacist for some drug to cure a constipation.

Not all false friends are "false cognates". Many of those which cause problems between English and other European languages are real cognates - words in different languages with similar forms sharing a common etymology - where one has undergone a semantic shift and the other has not. For example, the English adjective "actual", meaning "real", in contrast with imaginary or predicted, has true cognates in most Western European languages - e.g. French actuel, German aktuel, Italian attuale, Dutch actueel - all of which mean "current", of the moment. In this case it was the English word which changed its meaning in or around the eighteenth century. Eventually (after a long delay) and its cognates eventuellement, eventualmente, eventueel etc. (possibly, maybe) are another common example along the same lines.

Actual false cognates - homonyms or near-homonyms in different languages with distinct meanings and unrelated etymologies - are somewhat rarer, at least in fairly closely related languages like those of Europe (apart from a few non-indo-European isolates). Examples might be the (unlikely) confusion between butter and donkeys in Italian and Spanish (es burro = it asino; it burro = es mantequilla), or an English kiss and a Finnish cat, or French and English meanings of chair.

False friends are words from different languages which meaning differ
although they look almost the same. There are hundreds or thousands of
false friends, or faux amis, between English and French
(Actually, the countries themselves are faux amis...) Here is a
short list, with strong English words and
underlined French words. You may add new lists for French or other
languages.

actuellement vs actually: French people
will often use actually when they really mean at
the present time, because it's what actuellement means in
French. Actually may be translated in French with en
réalité. In a similar way, éventuellement means possibly,
while the French for eventually is
finalement.

when French people attendent, they are waiting for something,
and not attending a meeting.

French armes are always weapons. The word for the part of the
body that extends from the shoulder to the hand (arms)
is bras, which is another faux ami...

If an Englishman blesses you, you may go to heaven. If a Frenchman blesse, you may go to the hospital, because he has hurt you.

a French car is an intercity bus. It's also a conjunction
which means because. The French for car is
voiture.

caractère vs character: a caractère
refers to the temperament or characteristics of a person or a thing, not a
movie character (who is a personnage).

In France, you buy books in librairies and read them for free
in bibliothèques (publiques). In England, you buy books in
bookshops and read them for free in (public)
libraries.

A mail in France used to be an alley or a square planted with
trees. Nowadays, it is an equivalent of e-mail.

A French phrase is a sentence, while an English
phrase is only un groupe de mots or une
expression. But a French sentence is either a judgment
pronounced by a judge, or a statement that says seemingly profound things
about the people or the world.

a Frenchman who reste (stays) for a few days does not always
rest (se reposer).

French people are surprised when they see forsale in a shop, because sale means dirty and
fort (with a trailing t) means very.

a square in France is a small square which
really has the shape of a square (and which is usually
planted with trees in the middle). In London, Trafalgar square is
anything but square. Apparently France borrowed the word
from English and never modified its original meaning while it acquired new
meanings in England. This may be due to the fact that it's less commonly
used than the generic French name of a square:
place.

If a French employer wants to hire you for a stage, you may
accept if you are looking for an internship, not if you are an
actor and want to play on the stage (scène).

last but not least: if a Frenchman asks for les waters, he
doesn't want to drink water or to meet the family of a
pop star, but to go to the toilets (water-closet)!

Most of these false friends occur when one language borrows a
word from another language because it's interested in only one of the meanings
of this word. They may also result from random evolution of non-cognate
words.

Nowadays English meanings tend to penetrate the French vocabulary
because of American influence. For example, it is not uncommon to hear
décade instead of décennie, or réaliser with the
meaning of realize. I have even heard dramatique
used for positive changes, i.e with the English meaning.

Thanks Albert Herring for telling me about the difference between bookshops and bookstores...